The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties
- 1 March 1973
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 10 (1) , 84-99
- https://doi.org/10.2307/3212497
Abstract
The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.Keywords
This publication has 10 references indexed in Scilit:
- Truncation procedures for non-negative matricesJournal of Applied Probability, 1971
- Finite approximations to infinite non-negative matrices, II: refinements and applicationsMathematical Proceedings of the Cambridge Philosophical Society, 1968
- Stationary equations in continuous time Markov chainsTransactions of the American Mathematical Society, 1963
- On non-dissipative Markov chainsMathematical Proceedings of the Cambridge Philosophical Society, 1957
- The calculation of the ergodic projection for Markov chains and processes with a countable infinity of statesActa Mathematica, 1957
- Denumerable Markov processes and the associated contraction semigroups on lActa Mathematica, 1957
- SOME FURTHER PATHOLOGICAL EXAMPLES IN THE THEORY OF DENUMERABLE MARKOV PROCESSESThe Quarterly Journal of Mathematics, 1956
- A solution to a set of fundamental equations in Markov chainsProceedings of the American Mathematical Society, 1954
- On non-dissipative Markoff chains with an enumerable infinity of statesMathematical Proceedings of the Cambridge Philosophical Society, 1951
- Markoff chains with an enumerable number of states and a class of cascade processesMathematical Proceedings of the Cambridge Philosophical Society, 1951